{"id":947,"date":"2010-10-07T13:58:28","date_gmt":"2010-10-07T13:58:28","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=947"},"modified":"2010-10-07T13:58:28","modified_gmt":"2010-10-07T13:58:28","slug":"zero-is-not-a-local-ring","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=947","title":{"rendered":"Zero is not a local ring"},"content":{"rendered":"<p>Let R be a ring such that for every x in R either x or 1 &#8211; x is invertible. Then I claim that R is a local ring. Take some time to think this through&#8230;<\/p>\n<p>Brian Conrad complained <a href=\"http:\/\/math216.wordpress.com\/2010\/09\/25\/third-notes-affine-schemes\/#comment-146\">here<\/a> that the statement above is not true because the zero ring is not a local ring. I agree with him. The same mistake was made in the stacks project! Argh!<\/p>\n<p>Fixing it led me to review the definition of a locally ringed topos. I want the definition of a locally ringed topos (see Definition <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04EU\">Tag 04EU<\/a>) when applied to a ringed space to produce a locally ringed space. Hence I decided to add a condition that guarantees that 1 is &#8220;nowhere&#8221; 0 on a locally ringed topos. Any complaints?<\/p>\n<p>Note that Exercise 13.9 of Exposee IV in SGA IV suffers from the same confusion too (although, of course, I may be misreading it). I also haven&#8217;t read Hakim&#8217;s thesis which SGA tells you to do (my bad). Have you?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let R be a ring such that for every x in R either x or 1 &#8211; x is invertible. Then I claim that R is a local ring. Take some time to think this through&#8230; Brian Conrad complained here &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=947\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-947","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/947","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=947"}],"version-history":[{"count":6,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/947\/revisions"}],"predecessor-version":[{"id":953,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/947\/revisions\/953"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=947"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}